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Directory:Jon Awbrey/Papers/Differential Logic and Dynamic Systems 2.0 (view source)

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====Differential Forms====
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The actions of the difference operator <math>\mathrm{D}\!</math> and the tangent operator <math>\mathrm{d}\!</math> on the 16 bivariate propositions are shown in Tables&nbsp;A7 and A8.
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Table A7 expands the differential forms that result over a ''logical basis'':
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{| align="center" cellpadding="6" style="text-align:center"
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|
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<math>\{~ \texttt{(}\mathrm{d}x\texttt{)(}\mathrm{d}y\texttt{)}, ~\mathrm{d}x~\texttt{(}\mathrm{d}y\texttt{)}, ~\texttt{(}\mathrm{d}x\texttt{)}~\mathrm{d}y, ~\mathrm{d}x~\mathrm{d}y ~\}.\!</math>
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|}
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This set consists of the singular propositions in the first order differential variables, indicating mutually exclusive and exhaustive ''cells'' of the tangent universe of discourse.  Accordingly, this set of differential propositions may also be referred to as the cell-basis, point-basis, or singular differential basis.  In this setting it is frequently convenient to use the following abbreviations:
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{| align="center" cellpadding="6" style="text-align:center"
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|
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<math>\partial x ~=~ \mathrm{d}x~\texttt{(}\mathrm{d}y\texttt{)}\!</math> &nbsp; &nbsp; and &nbsp; &nbsp; <math>\partial y ~=~ \texttt{(}\mathrm{d}x\texttt{)}~\mathrm{d}y.\!</math>
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|}
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Table A8 expands the differential forms that result over an ''algebraic basis'':
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{| align="center" cellpadding="6" style="text-align:center"
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| <math>\{~ 1, ~\mathrm{d}x, ~\mathrm{d}y, ~\mathrm{d}x~\mathrm{d}y ~\}.\!</math>
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|}
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This set consists of the ''positive propositions'' in the first order differential variables, indicating overlapping positive regions of the tangent universe of discourse.  Accordingly, this set of differential propositions may also be referred to as the ''positive differential basis''.
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====Table A7. Differential Forms Expanded on a Logical Basis====
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<br>
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{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"
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|+ style="height:30px" | <math>\text{Table A7.} ~~ \text{Differential Forms Expanded on a Logical Basis}\!</math>
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|- style="background:ghostwhite; height:40px"
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| &nbsp;
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| style="border-right:none" | <math>f\!</math>
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| style="border-left:4px double black" | <math>\mathrm{D}f</math>
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| <math>\mathrm{d}f</math>
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|-
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| <math>f_{0}\!</math>
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| style="border-right:none" | <math>\texttt{(~)}\!</math>
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| style="border-left:4px double black" | <math>0\!</math>
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| <math>0\!</math>
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|-
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| <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math>
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| style="border-right:none" |
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<math>\begin{matrix}
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\texttt{(} x \texttt{)(} y \texttt{)}
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\\
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\texttt{(} x \texttt{)~} y \texttt{~}
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\\
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\texttt{~} x \texttt{~(} y \texttt{)}
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\\
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\texttt{~} x \texttt{~~} y \texttt{~}
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\end{matrix}</math>
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| style="border-left:4px double black" |
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<math>\begin{matrix}
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\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}
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& + &
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\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y
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& + &
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\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y
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\\
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y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}
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& + &
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\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y
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& + &
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\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y
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\\
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\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}
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& + &
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x & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y
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& + &
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\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y
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\\
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y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}
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& + &
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x & \texttt{(} \mathrm{d}x) ~ \mathrm{d}y
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& + &
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\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y
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\end{matrix}</math>
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|
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<math>\begin{matrix}
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\texttt{(} y \texttt{)} ~\partial x
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& + &
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\texttt{(} x \texttt{)} ~\partial y
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\\
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\texttt{~} y \texttt{~} ~\partial x
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& + &
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\texttt{(} x \texttt{)} ~\partial y
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\\
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\texttt{(} y \texttt{)} ~\partial x
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& + &
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\texttt{~} x \texttt{~} ~\partial y
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\\
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\texttt{~} y \texttt{~} ~\partial x
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& + &
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\texttt{~} x \texttt{~} ~\partial y
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\end{matrix}</math>
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|-
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| <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math>
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| style="border-right:none" |
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<math>\begin{matrix}
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\texttt{(} x \texttt{)}
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\\
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\texttt{~} x \texttt{~}
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\end{matrix}</math>
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| style="border-left:4px double black" |
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<math>\begin{matrix}
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\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \mathrm{d}x ~ \mathrm{d}y
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\\
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\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \mathrm{d}x ~ \mathrm{d}y
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\end{matrix}\!</math>
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|
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<math>\begin{matrix}
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\partial x
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\\
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\partial x
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\end{matrix}</math>
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|-
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| <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math>
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| style="border-right:none" |
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<math>\begin{matrix}
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\texttt{~(} x \texttt{,~} y \texttt{)~}
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\\
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\texttt{((} x \texttt{,~} y \texttt{))}
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\end{matrix}</math>
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| style="border-left:4px double black" |
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<math>\begin{matrix}
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\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y
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\\
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\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y
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\end{matrix}</math>
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|
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<math>\begin{matrix}
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\partial x & + & \partial y
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\\
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\partial x & + & \partial y
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\end{matrix}</math>
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|-
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| <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math>
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| style="border-right:none" |
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<math>\begin{matrix}
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\texttt{(} y \texttt{)}
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\\
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\texttt{~} y \texttt{~}
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\end{matrix}</math>
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| style="border-left:4px double black" |
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<math>\begin{matrix}
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\texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y & + & \mathrm{d}x ~ \mathrm{d}y
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\\
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\texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y & + & \mathrm{d}x ~ \mathrm{d}y
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\end{matrix}</math>
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|
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<math>\begin{matrix}
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\partial y
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\\
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\partial y
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\end{matrix}</math>
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|-
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| <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math>
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| style="border-right:none" |
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<math>\begin{matrix}
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\texttt{(~} x \texttt{~~} y \texttt{~)}
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\\
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\texttt{(~} x \texttt{~(} y \texttt{))}
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\\
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\texttt{((} x \texttt{)~} y \texttt{~)}
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\\
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\texttt{((} x \texttt{)(} y \texttt{))}
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\end{matrix}</math>
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| style="border-left:4px double black" |
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<math>\begin{matrix}
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y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}
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& + &
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x & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y
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& + &
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\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y
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\\
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\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}
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& + &
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x & \texttt{(} \mathrm{d}x) ~ \mathrm{d}y
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& + &
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\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y
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\\
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y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}
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& + &
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\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y
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& + &
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\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y
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\\
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\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}
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& + &
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\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y
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& + &
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\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y
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\end{matrix}</math>
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|
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<math>\begin{matrix}
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\texttt{~} y \texttt{~} ~\partial x
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& + &
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\texttt{~} x \texttt{~} ~\partial y
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\\
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\texttt{(} y \texttt{)} ~\partial x
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& + &
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\texttt{~} x \texttt{~} ~\partial y
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\\
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\texttt{~} y \texttt{~} ~\partial x
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& + &
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\texttt{(} x \texttt{)} ~\partial y
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\\
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\texttt{(} y \texttt{)} ~\partial x
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& + &
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\texttt{(} x \texttt{)} ~\partial y
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\end{matrix}</math>
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|-
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| <math>f_{15}\!</math>
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| style="border-right:none" | <math>\texttt{((~))}\!</math>
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| style="border-left:4px double black" | <math>0\!</math>
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| <math>0\!</math>
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|}
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<br>
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====Table A8. Differential Forms Expanded on an Algebraic Basis====
    
====Table A12. Detail of Calculation for the Difference Map====
 
====Table A12. Detail of Calculation for the Difference Map====
Jon Awbrey
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